If a very long indestructible object, such as a rod or spaceship, were to approach the event horizon of a black hole, the phenomenon known as gravitational time dilation would indeed cause significant differences in the passage of time for different parts of the object. This effect is known as tidal forces.
As the object gets closer to the black hole, the gravitational field it experiences becomes stronger. Near the event horizon, where the gravitational pull is extremely intense, the time dilation becomes more pronounced.
The part of the object closer to the black hole will experience a stronger gravitational field and, consequently, more time dilation. This means that time will pass more slowly for the section of the object closer to the black hole compared to the section farther away.
From an external observer's perspective, as the object approaches the event horizon, the differences in time dilation would become increasingly noticeable. The section of the object closer to the black hole would appear to slow down relative to the section farther away.
However, it's important to note that as the object approaches the event horizon, the gravitational tidal forces can become incredibly strong. These forces can cause significant stretching and deformation of the object, potentially tearing it apart. This phenomenon is known as spaghettification.
If the object is long enough and the tidal forces are extreme, the stretching can be so severe that the object may ultimately break apart before reaching the event horizon. The exact outcome would depend on the specific properties and strength of the object.
In summary, if a very long indestructible object were to approach the event horizon of a black hole, the part of the object closer to the black hole would experience more time dilation. However, the extreme tidal forces near the event horizon could lead to the object being torn apart before it reaches that point.